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This problem involves using trigonometry to find the horizontal distances to each ship and then calculating the distance between the two ships.

### Key Information:
- Height of the observer above sea level: \( h = 1000 \, \text{m} \)
- Angle of depression to Ship 1: \( \theta_1 = 12^\circ \)
- Angle of depression to Ship 2: \( \theta_2 = 10^\circ \)

### Step-by-Step Solution:
1. **Find the horizontal distance to Ship 1:**
   The angle of depression is below the horizontal, so we can use the tangent function:
   \[
   \tan(\theta_1) = \frac{h}{\text{distance to Ship 1 (d1)}}
   \]
   Solving for \( d_1 \):
   \[
   d_1 = \frac{h}{\tan(\theta_1)} = \frac{1000}{\tan(12^\circ)}
   \]

2. **Find the horizontal distance to Ship 2:**
   Similarly, for the second ship:
   \[
   \tan(\theta_2) = \frac{h}{\text{distance to Ship 2 (d2)}}
   \]
   Solving for \( d_2 \):
   \[
   d_2 = \frac{1000}{\tan(10^\circ)}
   \]

3. **Calculate the distance between the two ships:**
   The distance between the two ships is the difference between \( d_1 \) and \( d_2 \):
   \[
   \text{Distance between the two ships} = d_2 - d_1
   \]

I'll compute this now.The distance between the two ships is approximately \( 966.65 \, \text{m} \), which matches the given answer of 966 m.

Would you like more details on any part of the solution, or do you have any further questions?

Here are five related questions for you to explore:
1. How would the result change if the observer were at a different height?
2. What happens to the horizontal distances as the angles of depression increase?
3. Can you calculate the exact positions of the ships relative to the observer?
4. How would you approach this problem if the ships were not due east but at an angle to each other?
5. Can you apply a similar method to calculate distances in a 3D space?

**Tip:** Always ensure you convert angles from degrees to radians when working with trigonometric functions in most programming languages.

An observer on a cliff 1000 m above sea level sights two ships due east. The angles of depression (below the horizontal) of the ships are 10° and 12°. Find the distance between the two ships.

Learn how to calculate the distance between two ships viewed from a cliff 1000 meters above sea level. The angles of depression are 10° and 12°, and the distance is determined using trigonometric principles. Suitable for high school students (Grades 9-12).

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